On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization

Constraint qualifications in nonsmooth multiobjective optimization problem

“…Hence it is easy to see that the reverse inclusions in (3.4) and (3.5) are not satisfied atx = 0 for any of the indices 1 or 2. Therefore on the lines of Burachik and Rizvi [2], in order to obtain necessary conditions for a feasible solution of (MOP) to be an efficient/weak efficient solution, we assume the following:…”

“…Rizvi et al [12] then derived strong KKT conditions for an efficient solution of a differentiable programming problem using (GGCQ). Recently Burachik and Rizvi [2] used the M i sets introduced by Rizvi et al [12] to define regularity conditions named as Guignard regularity condition (GRC) and generalized Abadie regularity condition (GARC). Weak KKT conditions for an efficient solution using (GRC) and strong KKT conditions for Geoffrion proper efficient solution using (GARC) are obtained in [2] for a multiobjective programming problem with only inequality constraints where the functions are assumed to be continuously differentiable.…”

“…Motivated by the above research work, in this paper we define nonsmooth versions of (GRC) and (GARC) introduced by Burachik and Rizvi [2]. These newly defined versions are called generalized Guignard constraint qualification (GGCQ) and generalized Abadie constraint qualification (GACQ) respectively.…”

Locally Lipschitz vector optimization problems: second-order constraint qualifications, regularity condition and KKT necessary optimality conditions

“…In order to obtain these optimality conditions, constraint qualifications and regularity conditions are indispensable; see, for example, [12][13][14][15][16][17][18][19][20]. We recall here that these assumptions are called constraint qualifications (CQ) when they have to be fulfilled by the constraints of the problem, and they are called regularity conditions (RC) when they have to be fulfilled by both the objectives and the constraints of the problem; see [21] for more details.…”

“…Consider the following problem:min f (x) := (f 1 (x), f 2 (x)) subject to x ∈ Q 0 := {x ∈ R 2 | g(x) 0}, where f 1 (x) := |x 1 | + x 2 2 , f 2 (x) := −f 1 (x), g(x) := x 2 for all x = (x 1 , x 2 ) ∈ R 2 .Clearly,x = (0, 0) is a Geoffrion properly efficient solution. The optimality conditions of Burachik et al[21, Theorem 4.3] cannot be used for this problem as the functions f 1 and f 2 are not differentiable atx.…”